The most important thing, I think, is to do everything you possibly can to make sure you're not in immediate danger. The most immediately dangerous enemies are jetpack bots (because they move very quickly and can be unpredictable) and Mk 3 hammer bots (because they move relatively quickly and are difficult to dodge); archers are only a significant factor if you turn your back to them, sword bots can be dealt with almost at your leisure (they don't have much range, can be easily dodged, and can be easily taken out by hitting their flanks), and lower tier hammer bots are relatively simple to avoid (as their attack animation is quite long, giving you plenty of time to either dodge or hit them first).
With that in mind, the best upgrade path (in my opinion) is as follows:
1. Sword upgrades (Deflect Arrows 1 [to remove the threat of archers], Flame 1 [for the improved PTK], Deflect Arrows 2-3, Flame 2)
2, Kick + Get Up (the latter being the real goal; reducing your vulnerability when knocked over is extremely important in later levels)
3. Clone (for a second chance; replace when used)
4. Jetpack + Energy [regeneration, when available] (the jetpack allows you to much more safely attack several enemies, most notably Spider-tron 6000 and the Mk 3 hammer bot)
5. Fire Breath (not the best of upgrades [its charge time is longer than its active time with one energy bar], but can bypass swords if you don't want to get close and can hit Spider-tron 6000 without cutting its legs)
6. Bow Upgrades (fairly lackluster in the late-game, since most enemies' movements are impossible to predict at any meaningful range, but can be useful for taking out stationary targets. Aim Time is, in my opinion, just a waste of energy if you have any experience with shooter games)
7. By this point, why not get the hammer upgrades? It's still probably not a good idea to use them, at least until you've cleared out the archers.